In probability theory and statistics, a median is described as the number separating the higher half of a sample, a population, or a probability distribution, from the lower half. Probability theory is the branch of Mathematics concerned with analysis of random phenomena Statistics is a mathematical science pertaining to the collection analysis interpretation or explanation and presentation of Data. In Probability theory and Statistics, a probability distribution identifies either the probability of each value of an unidentified Random variable The median of a finite list of numbers can be found by arranging all the observations from lowest value to highest value and picking the middle one. If there is an even number of observations, the median is not unique, so one often takes the mean of the two middle values. In Statistics, mean has two related meanings the Arithmetic mean (and is distinguished from the Geometric mean or Harmonic mean At most half the population have values less than the median and at most half have values greater than the median. If both groups contain less than half the population, then some of the population is exactly equal to the median. For example, if a < b < c, then the median of the list {a, b, c} is b, and if a < b < c < d, then the median of the list {a, b, c, d} is the mean of b and c, i. e. it is (b + c)/2.

Notation

The median of some variable is denoted either as or as

Popular explanation

The difference between the median and the mean is illustrated in this simple example:

Suppose 19 paupers and 1 billionaire are in a room. Everyone removes all the money from their pockets and puts it on a table. Each pauper puts $5 on the table; the billionaire puts $1 billion (i. e. $10⁹) there. The total is then $1,000,000,095. If that money is divided equally among the 20 people, each gets $50,000,004. 75. That amount is the mean amount of money that the 20 people brought into the room. In Statistics, mean has two related meanings the Arithmetic mean (and is distinguished from the Geometric mean or Harmonic mean But the median amount is $5, since one may divide the group into two groups of 10 people each, and say that everyone in the first group brought in no more than $5, and each person in the second group brought in no less than $5. In a sense, the median is the amount that the typical person brought in. By contrast, the mean is not at all typical, since nobody in the room brought in an amount approximating $50,000,004. 75.

Measures of statistical dispersion

When the median is used as a location parameter in descriptive statistics, there are several choices for a measure of variability: the range, the interquartile range, the mean absolute deviation, and the median absolute deviation. In Statistics, a location family is a class of probability distributions parametrized by a scalar- or vector-valued parameter μ, which determines the "location" In Descriptive statistics, the range is the length of the smallest interval which contains all the data In Descriptive statistics, the interquartile range (IQR, also called the midspread, middle fifty and middle of the #s, is a measure of In Statistics, the absolute deviation of an element of a Data set is the absolute difference between that element and a given point In Statistics, the median absolute deviation (or " MAD " is a resistant measure of the variability of a univariate sample Since the median is the same as the second quartile, its calculation is illustrated in the article on quartiles. In Descriptive statistics, a quartile is any of the three values which divide the sorted Data set into four equal parts so that each part represents one fourth of

Working with computers, a population of integers should have an integer median. Thus, for an integer population with an even number of elements, there are two medians known as lower median and upper median. For floating point population, the median lies somewhere between the two middle elements, depending on the distribution. Median is the middle most value after arranging data by any order

Medians of probability distributions

For any probability distribution on the real line with cumulative distribution function F, regardless of whether it is any kind of continuous probability distribution, in particular an absolutely continuous distribution (and therefore has a probability density function), or a discrete probability distribution, a median m satisfies the inequalities

or

in which a Riemann-Stieltjes integral is used. In Probability theory and Statistics, a probability distribution identifies either the probability of each value of an unidentified Random variable In Mathematics, the real numbers may be described informally in several different ways In Probability theory and Statistics, the cumulative distribution function (CDF, also probability distribution function or just distribution function In Mathematics, one may talk about absolute continuity of functions and absolute continuity of measures, and these two notions are closely connected In Mathematics, a probability density function (pdf is a function that represents a Probability distribution in terms of Integrals Formally a probability In Mathematics, the Riemann-Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes For an absolutely continuous probability distribution with probability density function f, we have

Medians of particular distributions: The medians of certain types of distributions can be easily estimated from their parameters: The median of a normal distribution with mean μ and variance σ² is μ. In Mathematics, a probability density function (pdf is a function that represents a Probability distribution in terms of Integrals Formally a probability The normal distribution, also called the Gaussian distribution, is an important family of Continuous probability distributions applicable in many fields In fact, for a normal distribution, mean = median = mode. The median of a uniform distribution in the interval [a, b] is (a + b) / 2, which is also the mean. The median of a Cauchy distribution with location parameter x₀ and scale parameter y is x₀, the location parameter. The Cauchy–Lorentz distribution, named after Augustin Cauchy and Hendrik Lorentz, is a continuous Probability distribution. The median of an exponential distribution with rate parameter λ is the natural log of 2 divided by the rate parameter: ln2 / λ. WikipediaWikiProject Probability#Standards for a discussionof standards used for probability distribution articles such as this one In Probability theory and Statistics, a scale parameter is a special kind of Numerical parameter of a Parametric family of Probability distributions The median of a Weibull distribution with shape parameter k and scale parameter λ is λ(ln2)^{1 / k}. In Probability theory and Statistics, the Weibull distribution (named after Waloddi Weibull) is a continuous Probability distribution.

Medians in descriptive statistics

The median is primarily used for skewed distributions, which it represents differently than the arithmetic mean. In Probability theory and Statistics, skewness is a measure of the asymmetry of the Probability distribution of a real -valued In Mathematics and Statistics, the arithmetic Mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided Consider the multiset { 1, 2, 2, 2, 3, 9 }. In Mathematics, a multiset (or bag) is a generalization of a set. The median is 2 in this case, as is the mode, and it might be seen as a better indication of central tendency than the arithmetic mean of 3. In Statistics, the mode is the value that occurs the most frequently in a Data set or a Probability distribution. In Mathematics, an average, or central tendency of a Data set refers to a measure of the "middle" or " expected " value of In Mathematics and Statistics, the arithmetic Mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided 166….

Calculation of medians is a popular technique in summary statistics and summarizing statistical data, since it is simple to understand and easy to calculate, while also giving a measure that is more robust in the presence of outlier values than is the mean. In Descriptive statistics, summary statistics are used to summarize a set of observations in order to communicate as much as possible as simply as possible Descriptive Statistics are used to describe the basic features of the Data gathered from an experimental study in various ways In Statistics, an outlier is an observation that is numerically distant from the rest of the data. In Statistics, mean has two related meanings the Arithmetic mean (and is distinguished from the Geometric mean or Harmonic mean

Theoretical properties

An optimality property

The median is also the central point which minimizes the average of the absolute deviations; in the example above this would be (1 + 0 + 0 + 0 + 1 + 7) / 6 = 1. 5 using the median, while it would be 1. 944 using the mean. In the language of probability theory, the value of c that minimizes

is the median of the probability distribution of the random variable X. A random variable is a rigorously defined mathematical entity used mainly to describe Chance and Probability in a mathematical way Note, however, that c is not always unique, and therefore not well defined in general.

An inequality relating means and medians

For continuous probability distributions, the difference between the median and the mean is less than or equal to one standard deviation. In Probability and Statistics, the standard deviation is a measure of the dispersion of a collection of values See an inequality on location and scale parameters. For Probability distributions having an Expected value and a Median, the mean (i

Efficient computation

Even though sorting n items takes in general O(n log n) operations, by using a "divide and conquer" algorithm the median of n items can be computed with only O(n) operations (in fact, you can always find the k-th element of a list of values with this method; this is called the selection problem). In Computer science and Mathematics, a sorting algorithm is an Algorithm that puts elements of a list in a certain order. In mathematics big O notation (so called because it uses the symbol O) describes the limiting behavior of a function for very small or very large arguments In Computer science, divide and conquer ( D&C) is an important Algorithm design Paradigm. In mathematics big O notation (so called because it uses the symbol O) describes the limiting behavior of a function for very small or very large arguments In Computer science, a selection algorithm is an Algorithm for finding the k th smallest number in a list called Order statistics '

Easy explanation (Statistics)

As an example, we will calculate the median of the following population of numbers: 1, 5, 2, 8, 7.

Start by sorting the numbers: 1, 2, 5, 7, 8.

In this case, 5 is the median, because when the numbers are sorted, it is the middle number. If there is an even amount of numbers, the median is the arithmetic mean of the two middle numbers. In Mathematics and Statistics, the arithmetic Mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided

See also

• Order statistic
• An inequality on location and scale parameters
• The median is the 2nd quartile, 5th decile, and 50th percentile. In Statistics, the k th order statistic of a Statistical sample is equal to its k th-smallest value For Probability distributions having an Expected value and a Median, the mean (i In Descriptive statistics, a quartile is any of the three values which divide the sorted Data set into four equal parts so that each part represents one fourth of A percentile is the value of a variable below which a certain percent of observations fall
• Median voter theory
• The median in general is a biased estimator. The median voter theory, also known as the median voter theorem and the median voter model, is a famous voting model positing that in a Majority In Statistics, the difference between an Estimator 's Expected value and the true value of the parameter being estimated is called the bias.
• Median graph
• The centerpoint is a generalization of the median for data in higher dimensions. In Mathematics, and more specifically Graph theory, a median graph is an Undirected graph in which any three vertices a, b In Statistics and Computational geometry, the centerpoint is a generalization of the Median to data in two or more dimensions

External links

• A Guide to Understanding & Calculating the Median

• Median as a weighted arithmetic mean of all Sample Observations

• On-line calculator

• Calculating the median

• A problem involving the mean, the median, and the mode.

• mathworld: Statistical Median

This article incorporates material from Median of a distribution on PlanetMath, which is licensed under the GFDL. PlanetMath is a free, collaborative online Mathematics Encyclopedia.